Stieltjes constants appearing in the Laurent expansion of the hyperharmonic zeta function

Abstract

In this paper, we consider meromorphic extension of the function \[ ζ_{h^{\left( r\right) }}\left( s\right) =\sum_{k=1}^{\infty} \frac{h_{k}^{\left( r\right) }}{k^{s}},\text{ }\operatorname{Re}\left( s\right) >r, \] (which we call \textit{hyperharmonic zeta function}) where $h_n^{(r)}$ are the hyperharmonic numbers. We establish certain constants, denoted ${\gamma }_{h^{\left(r\right)}}\left(m\right)$, which naturally occur in the Laurent expansion of ${\zeta }_{h^{\left(r\right)}}\left(s\right)$. Moreover, we show that the constants ${\gamma }_{h^{\left(r\right)}}\left(m\right)$ and integrals involving generalized exponential integral can be written as a finite combination of some special constants.

Cited by in corpus (1)

• On the Stieltjes constants with respect to harmonic zeta functions